# Proofs of trigonometric identities i, sin 2x = 2sin x cos x

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In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "*x* = *x*" or usefully true, such as the Pythagorean Theorem"s "*a*2 + *b*2 = *c*2" for right triangles. There are loads of trigonometric identities, but the following are the ones you"re most likely khổng lồ see và use.

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Basic & Pythagorean, Angle-Sum & -Difference, Double-Angle, Half-Angle, Sum, Product

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact lớn help you keep straight that cosecant goes with sine và secant goes with cosine.

The following (particularly the first of the three below) are called "Pythagorean" identities.

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Note that the three identities above all involve squaring and the number 1. You can see the Pythagorean-Thereom relationship clearly if you consider the unit circle, where the angle is *t*, the "opposite" side is sin(*t*) = *y*, the "adjacent" side is cos(*t*) = *x*, & the hypotenuse is 1.

We have additional identities related to lớn the functional status of the trig ratios:

Notice in particular that sine and tangent are odd functions, being symmetric about the origin, while cosine is an even function, being symmetric about the *y*-axis. The fact that you can take the argument"s "minus" sign outside (for sine và tangent) or eliminate it entirely (forcosine) can be helpful when working with complicated expressions.

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*Angle-Sum và -Difference Identities*

sin(α + β) = sin(α) cos(β) + cos(α) sin(β)

sin(α − β) = sin(α) cos(β) − cos(α) sin(β)

cos(α + β) = cos(α) cos(β) − sin(α) sin(β)

cos(α − β) = cos(α) cos(β) + sin(α) sin(β)

/ <1 - tan(a)tan(b)>, tan(a - b) =

By the way, in the above identities, the angles are denoted by Greek letters. The a-type letter, "α", is called "alpha", which is pronounced "AL-fuh". The b-type letter, "β", is called "beta", which is pronounced "BAY-tuh".

sin(2*x*) = 2 sin(*x*) cos(*x*)

cos(2*x*) = cos2(*x*) − sin2(*x*) = 1 − 2 sin2(*x*) = 2 cos2(*x*) − 1

/ <1 - tan^2(x)>">

, cos(x/2) = +/- sqrt<(1 + cos(x))/2>, tan(x/2) = +/- sqrt<(1 - cos(x))/(1 + cos(x))>" style="min-width:398px;">

The above identities can be re-stated by squaring each side and doubling all of the angle measures. The results are as follows:

You will be using all of these identities, or nearly so, for proving other trig identities & for solving trig equations. However, if you"re going on lớn study calculus, pay particular attention to the restated sine và cosine half-angle identities, because you"ll be using them a *lot* in integral calculus.